Mechanisms

Horizon Collapse

Where cooperation stops being rational

Cooperation in a repeated game is held up by the shadow of the future. Make the endpoint visible and that shadow shortens: at the round where the discounted value of what remains falls below the Folk Theorem threshold, defection becomes the rational move — and because both sides can compute that round, the unravelling runs backwards from the end to the present.

The Folk Theorem says that patience plus an unbounded future can sustain cooperation as an equilibrium. The clause that gets dropped is unbounded. A season end, a vesting cliff, an auction window, an agent task-loop stop — any visible endpoint converts an indefinitely repeated game into a finite one, and backward induction does the rest.

This is why cooperation in agent systems often fails suddenly rather than gradually. Nothing degrades; the arithmetic simply crosses a threshold, and the same agent that cooperated a thousand times defects on the run where the remaining horizon got too short to pay for cooperating.

The fix is a rolling horizon: remove the visible endpoint so the remaining future never shrinks below the threshold. The linter below computes the exact round where your game crosses it, and lets you toggle the rolling-horizon clause to see the collapse disappear.

Where this is worked

• Issue 5: Why Threats Work — The Neural Economics of Cooperation

The tool

The Horizon-Collapse Linter — free, runs entirely in your browser, nothing is fetched and nothing is stored.

Set the rounds, the discount factor δ and the named endpoints; it returns the first round where δ^remaining drops below the Folk Theorem threshold and cooperation unravels. Basis: Fudenberg & Maskin (1986); Axelrod (1984).